Properties

Label 2-4840-1.1-c1-0-93
Degree $2$
Conductor $4840$
Sign $-1$
Analytic cond. $38.6475$
Root an. cond. $6.21671$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.34·3-s − 5-s − 2.86·7-s + 2.51·9-s − 6.24·13-s − 2.34·15-s + 3.73·17-s + 8.24·19-s − 6.73·21-s + 8.94·23-s + 25-s − 1.13·27-s − 8.69·29-s − 5.21·31-s + 2.86·35-s − 2.69·37-s − 14.6·39-s − 8.21·41-s − 7.13·43-s − 2.51·45-s + 1.31·47-s + 1.21·49-s + 8.76·51-s − 0.180·53-s + 19.3·57-s + 3.55·59-s − 5.94·61-s + ⋯
L(s)  = 1  + 1.35·3-s − 0.447·5-s − 1.08·7-s + 0.839·9-s − 1.73·13-s − 0.606·15-s + 0.905·17-s + 1.89·19-s − 1.46·21-s + 1.86·23-s + 0.200·25-s − 0.218·27-s − 1.61·29-s − 0.936·31-s + 0.484·35-s − 0.443·37-s − 2.35·39-s − 1.28·41-s − 1.08·43-s − 0.375·45-s + 0.191·47-s + 0.173·49-s + 1.22·51-s − 0.0247·53-s + 2.56·57-s + 0.462·59-s − 0.761·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4840\)    =    \(2^{3} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(38.6475\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4840,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good3 \( 1 - 2.34T + 3T^{2} \)
7 \( 1 + 2.86T + 7T^{2} \)
13 \( 1 + 6.24T + 13T^{2} \)
17 \( 1 - 3.73T + 17T^{2} \)
19 \( 1 - 8.24T + 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 + 8.69T + 29T^{2} \)
31 \( 1 + 5.21T + 31T^{2} \)
37 \( 1 + 2.69T + 37T^{2} \)
41 \( 1 + 8.21T + 41T^{2} \)
43 \( 1 + 7.13T + 43T^{2} \)
47 \( 1 - 1.31T + 47T^{2} \)
53 \( 1 + 0.180T + 53T^{2} \)
59 \( 1 - 3.55T + 59T^{2} \)
61 \( 1 + 5.94T + 61T^{2} \)
67 \( 1 - 2.27T + 67T^{2} \)
71 \( 1 + 2.51T + 71T^{2} \)
73 \( 1 + 8.36T + 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 - 3.55T + 83T^{2} \)
89 \( 1 - 0.732T + 89T^{2} \)
97 \( 1 - 1.75T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.74697492204849863402540221024, −7.31806567673389138715956571654, −6.92832993974583204882884648032, −5.50504968734959993011773122100, −5.03331867082391896971847480087, −3.75307303940093676519528750289, −3.17260247312698151805572666843, −2.81055263729598350810259345707, −1.54913927904250273522743298472, 0, 1.54913927904250273522743298472, 2.81055263729598350810259345707, 3.17260247312698151805572666843, 3.75307303940093676519528750289, 5.03331867082391896971847480087, 5.50504968734959993011773122100, 6.92832993974583204882884648032, 7.31806567673389138715956571654, 7.74697492204849863402540221024

Graph of the $Z$-function along the critical line